The Hayman--Wu constant is $π^2$

Abstract

We show that for every conformal map $ϕ:\mathbb{D}\toΩ\subsetneq\mathbb{C}$ from the unit disk onto a simply connected proper domain $Ω$, the length of $ϕ^{-1}(Ω\cap L)$ is at most $π^2$ for every line $L$.